1 Overview

Beyond the elicited informative prior, regulatory submissions typically require one or more alternative prior specifications to demonstrate robustness. bayprior provides three well-established alternatives:

Prior type	Function	Method	Reference
Robust mixture	`robust_prior()`	Mixes informative + vague	Schmidli et al. (2014)
Sceptical	`sceptical_prior()`	Centred at null effect	Spiegelhalter et al. (1994)
Power prior	`calibrate_power_prior()`	Down-weights historical data	Ibrahim & Chen (2000)

2 Robust Mixture Prior

2.1 Concept

A robust prior protects against prior misspecification by mixing the informative elicited prior with a vague (diffuse) component:

\[\pi_{\text{robust}}(\theta) = (1 - w) \cdot \pi_{\text{informative}}(\theta) + w \cdot \pi_{\text{vague}}(\theta)\]

The vague component — a wide Normal centred at the informative prior mean — ensures the posterior is never completely dominated by a conflicting prior. The default weight is \(w = 0.20\) (80% informative, 20% vague).

2.2 Usage

informative <- elicit_beta(
  mean      = 0.30,
  sd        = 0.08,
  method    = "moments",
  label     = "Response rate"
)

rob <- robust_prior(
  informative  = informative,
  vague_weight = 0.20,
  label        = "Robust mixture prior"
)
plot(rob)

cat("Informative component weight:", 1 - rob$vague_weight, "\n")
#> Informative component weight: 0.8
cat("Vague component weight:      ", rob$vague_weight, "\n")
#> Vague component weight:       0.2
cat("Mixture mean:", round(rob$fit_summary$mean, 4), "\n")
#> Mixture mean: 0.3
cat("Mixture SD:  ", round(rob$fit_summary$sd,   4), "\n")
#> Mixture SD:   0.3649

2.3 Varying the Vague Weight

A higher vague weight makes the prior more diffuse and reduces its influence on the posterior:

oldpar <- par(mfrow = c(1, 1))
weights <- c(0.10, 0.20, 0.30, 0.50)
cols    <- c("#185FA5", "#1D9E75", "#D85A30", "#888780")

x <- seq(0, 0.8, length.out = 300)
plot(x, bayprior:::.eval_density_vec(informative, x),
     type = "l", lwd = 2, col = "#185FA5",
     xlab = "Response rate", ylab = "Density",
     main = "Effect of vague weight on robust prior",
     ylim = c(0, 6))
for (i in seq_along(weights)) {
  r <- robust_prior(informative, vague_weight = weights[i])
  lines(x, bayprior:::.eval_density_vec(r, x),
        col = cols[i], lwd = 1.5, lty = i + 1)
}
legend("topright",
       legend = c("Informative",
                  paste0("w = ", weights)),
       col    = c("#185FA5", cols),
       lwd    = 2,
       lty    = c(1, 2, 3, 4, 5),
       bty    = "n", cex = 0.8)

par(oldpar)

2.4 Vague SD Multiplier

The vague component’s SD defaults to 10× the informative prior’s SD. Adjust with vague_sd:

rob_narrow <- robust_prior(informative, vague_weight = 0.20,
                           vague_sd = 2 * informative$fit_summary$sd)
rob_wide   <- robust_prior(informative, vague_weight = 0.20,
                           vague_sd = 20 * informative$fit_summary$sd)

cat("Narrow vague SD: ", round(rob_narrow$components$vague$fit_summary$sd, 3), "\n")
#> Narrow vague SD:  0.16
cat("Default vague SD:", round(rob$components$vague$fit_summary$sd, 3), "\n")
#> Default vague SD: 0.8
cat("Wide vague SD:   ", round(rob_wide$components$vague$fit_summary$sd, 3), "\n")
#> Wide vague SD:    1.6

3 Sceptical Prior

3.1 Concept

The sceptical prior (Spiegelhalter & Freedman, 1994) represents the view of a conservative regulator who is sceptical of a treatment effect. It is centred at the null value of the treatment effect with width calibrated to a chosen strength of scepticism.

This is the FDA’s recommended sensitivity prior for trials using informative priors: the trial conclusions should hold even under a prior that places most mass at “no effect.”

3.2 Normal Family (Mean Differences, Log Odds Ratios)

For continuous or log-scale quantities where the null is typically 0:

sc_weak <- sceptical_prior(
  null_value = 0, family = "normal", strength = "weak",
  label = "Log OR (weak sceptic)"
)
sc_moderate <- sceptical_prior(
  null_value = 0, family = "normal", strength = "moderate",
  label = "Log OR (moderate sceptic)"
)
sc_strong <- sceptical_prior(
  null_value = 0, family = "normal", strength = "strong",
  label = "Log OR (strong sceptic)"
)

cat("Weak SD:    ", sc_weak$fit_summary$sd, "\n")
#> Weak SD:     1
cat("Moderate SD:", sc_moderate$fit_summary$sd, "\n")
#> Moderate SD: 0.5
cat("Strong SD:  ", sc_strong$fit_summary$sd, "\n")
#> Strong SD:   0.25

plot(sc_moderate)

The SD mapping by strength:

Strength	SD	Interpretation
`weak`	1.0	Vague scepticism — wide prior around null
`moderate`	0.5	2-SD departure from null has ~5% prior probability
`strong`	0.25	Very concentrated at null — very sceptical

3.3 Beta Family (Response Rates)

For binary endpoints, null_value must be in \((0, 1)\) — it represents the null response rate, not a difference:

# Null response rate of 20%: sceptic believes treatment is no better than 20%
sc_beta <- sceptical_prior(
  null_value = 0.20,
  family     = "beta",
  strength   = "moderate",
  label      = "Response rate (sceptical)"
)
plot(sc_beta)

3.4 Log-Normal Family (Hazard Ratios)

For hazard ratios, the null is HR = 1, which corresponds to null_value = 0 on the log scale:

sc_hr <- sceptical_prior(
  null_value = 0,         # log(1) = 0, i.e. HR = 1
  family     = "lognormal",
  strength   = "moderate",
  label      = "Hazard ratio (sceptical)"
)
plot(sc_hr)

3.5 Enthusiastic vs Sceptical Pair

The FDA recommends presenting conclusions under both an enthusiastic prior (favouring treatment benefit) and a sceptical prior:

enthusiastic <- elicit_beta(
  mean = 0.45, sd = 0.08,
  method = "moments", label = "Response rate (enthusiastic)"
)
sceptical <- sceptical_prior(
  null_value = 0.20, family = "beta", strength = "moderate",
  label = "Response rate (sceptical)"
)

data_obs <- list(type = "binary", x = 18, n = 40)

post_enth <- bayprior:::.conjugate_update(enthusiastic, data_obs)
post_scep <- bayprior:::.conjugate_update(sceptical,    data_obs)

cat("Posterior mean (enthusiastic):", round(post_enth$fit_summary$mean, 3), "\n")
#> Posterior mean (enthusiastic): 0.45
cat("Posterior mean (sceptical):   ", round(post_scep$fit_summary$mean, 3), "\n")
#> Posterior mean (sceptical):    0.356
cat("Posterior SD (enthusiastic):  ", round(post_enth$fit_summary$sd,   3), "\n")
#> Posterior SD (enthusiastic):   0.056
cat("Posterior SD (sceptical):     ", round(post_scep$fit_summary$sd,   3), "\n")
#> Posterior SD (sceptical):      0.059

4 Power Prior

4.1 Concept

The power prior (Ibrahim & Chen, 2000) provides a principled method for incorporating historical data by down-weighting it by a factor \(\delta \in (0, 1]\):

\[\pi(\theta | D_0, \delta) \propto L(\theta | D_0)^\delta \cdot \pi_0(\theta)\]

where \(D_0\) is the historical data and \(\delta\) controls how much weight it receives. \(\delta = 1\) fully incorporates the historical data (standard Bayesian updating); \(\delta \to 0\) ignores it entirely.

4.2 Calibrating \(\delta\)

calibrate_power_prior() selects \(\delta\) to achieve a target Bayes Factor between the historical-data-informed prior and the current likelihood — ensuring the historical data is incorporated only to the extent it is compatible with current data:

base <- elicit_beta(
  mean   = 0.50,
  sd     = 0.20,
  method = "moments",
  label  = "Response rate"
)

calib <- calibrate_power_prior(
  historical_data = list(type = "binary", x = 12, n = 40),
  current_data    = list(type = "binary", x = 18, n = 50),
  base_prior      = base,
  target_bf       = 3,
  delta_grid      = seq(0.05, 1.0, by = 0.05),
  method          = "bayes_factor"
)
print(calib)

plot(calib)

The calibration curves show:

Top panel: Bayes Factor vs \(\delta\). The dashed red line is the target BF; the dotted green line marks the optimal \(\delta\).
Bottom panel: Box p-value vs \(\delta\). Values below 0.05 indicate conflict at that weight.

4.3 Compatibility Method

Alternatively, select \(\delta\) to be the largest value for which the historical-data-informed prior shows no conflict with the current data:

calib_compat <- calibrate_power_prior(
  historical_data = list(type = "binary", x = 12, n = 40),
  current_data    = list(type = "binary", x = 18, n = 50),
  base_prior      = base,
  method          = "compatibility",
  delta_grid      = seq(0.05, 1.0, by = 0.05)
)
cat("Optimal delta (BF method):           ", calib$delta_opt, "\n")
#> Optimal delta (BF method):            0.05
cat("Optimal delta (compatibility method):", calib_compat$delta_opt, "\n")
#> Optimal delta (compatibility method): 1

4.4 Power Prior for Other Families

Power prior updating is supported for Beta, Normal, Gamma, Log-Normal, and Mixture priors. For a Normal prior with continuous historical data:

base_norm <- elicit_normal(
  mean = 0.0, sd = 0.5,
  method = "moments", label = "Mean difference"
)

calib_norm <- calibrate_power_prior(
  historical_data = list(type = "continuous", x = 0.35, sd = 0.3, n = 60),
  current_data    = list(type = "continuous", x = 0.42, sd = 0.3, n = 80),
  base_prior      = base_norm,
  target_bf       = 3,
  delta_grid      = seq(0.05, 1.0, by = 0.10),
  method          = "bayes_factor"
)
print(calib_norm)

5 Choosing the Right Alternative Prior

library(knitr)
kable(data.frame(
  Situation = c(
    "No conflict, regulatory requirement",
    "Mild conflict detected",
    "Severe conflict detected",
    "Historical data available",
    "FDA enthusiastic/sceptical pair required"
  ),
  `Recommended prior` = c(
    "Robust mixture (w = 0.20)",
    "Robust mixture (w = 0.30-0.40)",
    "Sceptical prior (moderate-strong)",
    "Power prior (calibrated)",
    "Sceptical prior as second arm"
  ),
  check.names = FALSE
), align = "ll")

Situation	Recommended prior
No conflict, regulatory requirement	Robust mixture (w = 0.20)
Mild conflict detected	Robust mixture (w = 0.30-0.40)
Severe conflict detected	Sceptical prior (moderate-strong)
Historical data available	Power prior (calibrated)
FDA enthusiastic/sceptical pair required	Sceptical prior as second arm

6 Including Robust Priors in the Regulatory Report

All three robust prior types — robust mixture, sceptical, and power prior — are automatically included in the downloaded prior justification report when they have been computed in the session. Pass them directly to prior_report():

# After running the analyses above...
prior_report(
  prior           = prior,
  conflict        = cd,
  sensitivity     = sa,
  robust_prior    = rob,        # adds "Robust Mixture" section to report
  sceptical_prior = scep,       # adds "Sceptical Prior" section to report
  power_prior     = calib,      # adds "Power Prior" section with calibration table
  output_format   = "html",
  output_file     = "prior_justification_report",
  trial_name      = "TRIAL-001",
  sponsor         = "BioPharma Ltd",
  author          = "J. Smith, Biostatistician"
)

Each section in the report includes a parameter summary table and the corresponding density or calibration plot. The compliance checklist in the report automatically marks “Robust / sceptical prior computed” as Complete when any of the three types is supplied.

When using the Shiny app, the robust priors flow into the report automatically — simply run the analyses in the Robust Priors panel before clicking Download Report.

Note: prior_report() requires devtools::install(), not just devtools::load_all(). Quarto spawns a fresh R session that requires the package to be properly installed.

7 References

Ibrahim, J. G. & Chen, M.-H. (2000). Power prior distributions for regression models. Statistical Science, 15, 46–60.

Schmidli, H., Gsteiger, S., Roychoudhury, S., O’Hagan, A., Spiegelhalter, D., & Neuenschwander, B. (2014). Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics, 70, 1023–1032.

Spiegelhalter, D. J., Freedman, L. S., & Parmar, M. K. B. (1994). Bayesian approaches to randomized trials. Journal of the Royal Statistical Society A, 157, 357–416.

Gravestock, I. & Held, L. (2017). Adaptive power priors with empirical Bayes for clinical trials. Pharmaceutical Statistics, 16, 349–360.

Robust, Sceptical, and Power Priors

Ndoh Penn

2026-05-29